Ordinary differential equation - Revision history

Rongzhou: /* Reduction to a first-order system */

2026-10-10T19:53:00Z

Reduction to a first-order system

← Older revision		Revision as of 19:53, 10 October 2026
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	== Reduction to a first-order system ==		== Reduction to a first-order system ==

	For an ordinary differential equation, considering successive derivatives of the unknown function as new unknown functions ~~may reduce~~ the ~~order of~~ the original equation ~~without losing any information~~:		For an ordinary differential equation of any order higher than one, considering successive derivatives of the unknown function as new unknown functions reduces the the original equation to a system of first-order equations:

	{{#content:Q1991}}		{{#content:Q1991}}

Rongzhou: /* Linear equations and superposition */

2026-10-10T19:51:17Z

Linear equations and superposition

← Older revision		Revision as of 19:51, 10 October 2026
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	{{#content:Q1617}}		{{#content:Q1617}}

	~~== Linear equations and superposition ==~~

	~~An equation is '''linear''' when its terms in the unknown are of the first degree; collecting them on the left gives~~

	~~$$L[y]\equiv y^{(n)}+a_{n-1}(x)\,y^{(n-1)}+\cdots+a_{1}(x)\,y'+a_{0}(x)\,y=g(x)$$~~

	~~where the coefficients $a_{0}$, …, $a_{n-1}$ depend on $x$ only. The operator $L$ is linear, $L[c_{1}y_{1}+c_{2}y_{2}]=c_{1}L[y_{1}]+c_{2}L[y_{2}]$, so solutions combine:~~

	$$
	~~\begin{cases}~~
	~~L[y_{1}]=L[y_{2}]=0 \ \Longrightarrow\ L[c_{1}y_{1}+c_{2}y_{2}]=c_{1}\cdot 0+c_{2}\cdot 0=0 & \text{(superposition)} \\~~
	~~L[y_{h}]=0,\quad L[y_{p}]=g \ \Longrightarrow\ L[y_{h}+y_{p}]=0+g=g & \text{(homogeneous + particular)}~~
	~~\end{cases}~~
	$$

	Hence the general solution of a linear equation is $y=y_{h}+y_{p}$: the '''homogeneous solution''' $y_{h}$ carries the $n$ arbitrary constants, and $y_{p}$ is any one solution of the full equation. Neither rule survives for a nonlinear equation, such as $y'=y^{2}$:

	~~$$y_{1}'=y_{1}^{2},\quad y_{2}'=y_{2}^{2} \ \Longrightarrow\ (y_{1}+y_{2})'=y_{1}^{2}+y_{2}^{2}\neq (y_{1}+y_{2})^{2}$$~~

	~~so the sum of two solutions is not a solution.<ref>{{#cite:Q1981}}</ref>~~

	== See also ==		== See also ==

Rongzhou at 19:39, 10 October 2026

2026-10-10T19:39:27Z

← Older revision		Revision as of 19:39, 10 October 2026
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	'''An ordinary differential equation''' ('''ODE''') is a differential equation whose unknown function depends on a single independent variable~~. An equation whose unknown depends on several independent variables is a '''partial differential equation''' instead~~.		'''An ordinary differential equation''' ('''ODE''') is a differential equation whose unknown function depends on a single independent variable.

	== General form ==		== General form ==

Rongzhou: /* Solutions and initial conditions */

2026-10-10T19:38:37Z

Solutions and initial conditions

← Older revision		Revision as of 19:38, 10 October 2026
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	so the sum of two solutions is not a solution.<ref>{{#cite:Q1981}}</ref>		so the sum of two solutions is not a solution.<ref>{{#cite:Q1981}}</ref>

	~~== Solutions and initial conditions ==~~

	~~A '''solution''' is a function that satisfies the equation identically, as $\cos\omega_{0}t$ satisfies the harmonic oscillator equation:~~

	~~$$x=\cos\omega_{0}t \ \Longrightarrow\ \ddot{x}=-\omega_{0}^{2}\cos\omega_{0}t=-\omega_{0}^{2}x \ \Longrightarrow\ \ddot{x}+\omega_{0}^{2}x=0$$~~

	~~Together with $\sin\omega_{0}t$ it gives the general solution of that equation, with one arbitrary constant for each order:~~

	~~$$x(t)=C_{1}\cos\omega_{0}t+C_{2}\sin\omega_{0}t$$~~

	~~Fixing the constants requires as many conditions as the order. For a mass pulled to the position $x_{0}$ and released from rest at $t=0$,~~

	~~$$x(0)=C_{1}=x_{0} \ \Longrightarrow\ \dot{x}(t)=-C_{1}\omega_{0}\sin\omega_{0}t+C_{2}\omega_{0}\cos\omega_{0}t \ \Longrightarrow\ \dot{x}(0)=C_{2}\omega_{0}=0$$~~

	~~so $C_{2}=0$, and the particular solution is the oscillation~~

	~~$$x(t)=x_{0}\cos\omega_{0}t$$~~

	~~Imposed at a single point, the $n$ conditions~~

	~~$$y(x_{0})=y_{0},\qquad y'(x_{0})=y_{1},\qquad \ldots,\qquad y^{(n-1)}(x_{0})=y_{n-1}$$~~

	~~form an '''initial value problem'''; at the first order the single condition is~~

	~~{{#content:Q1582}}~~

	~~Conditions imposed at two different points are '''boundary conditions''', and the problem is a boundary value problem, which may have no solution or infinitely many.<ref>{{#cite:Q1982}}</ref>~~

	== See also ==		== See also ==

Rongzhou: /* Reduction to a first-order system */

2026-10-10T18:37:13Z

Reduction to a first-order system

← Older revision		Revision as of 18:37, 10 October 2026
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	which involves the second derivative.		which involves the second derivative.

	To reduce the equation to two first degree equations, introduce the angular acceleration $\omega=\frac{d\theta}{dt}$ as a second unknown:		To reduce the equation to a system of two first degree equations, introduce the angular acceleration $\omega=\frac{d\theta}{dt}$ as a second unknown:

	{{#content:Q1617}}		{{#content:Q1617}}

Rongzhou: /* Reduction to a first-order system */

2026-10-10T18:25:37Z

Reduction to a first-order system

← Older revision		Revision as of 18:25, 10 October 2026
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	which involves the second derivative.		which involves the second derivative.

	To reduce the equation to ~~the~~ first degree, introduce the angular ~~velocity~~ $\omega=\frac{d\theta}{dt}$ as a second unknown:		To reduce the equation to two first degree equations, introduce the angular acceleration $\omega=\frac{d\theta}{dt}$ as a second unknown:

	{{#content:Q1617}}		{{#content:Q1617}}

Rongzhou: /* Reduction to a first-order system */

2026-10-10T18:21:38Z

Reduction to a first-order system

← Older revision		Revision as of 18:21, 10 October 2026
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	which involves the second derivative.		which involves the second derivative.

	To reduce the equation to the first degree, introduce the angular velocity $w=\frac{d\theta}{dt}$ as a second unknown:		To reduce the equation to the first degree, introduce the angular velocity $\omega=\frac{d\theta}{dt}$ as a second unknown:

	{{#content:Q1617}}		{{#content:Q1617}}

Rongzhou: /* Reduction to a first-order system */

2026-10-10T18:11:49Z

Reduction to a first-order system

← Older revision		Revision as of 18:11, 10 October 2026
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	== Reduction to a first-order system ==		== Reduction to a first-order system ==

	For ~~any~~ ordinary differential equation, considering successive derivatives of the unknown function as new unknown functions may reduce the order of the original equation without losing any information:		For an ordinary differential equation, considering successive derivatives of the unknown function as new unknown functions may reduce the order of the original equation without losing any information:

	{{#content:Q1991}}		{{#content:Q1991}}

Rongzhou: /* Reduction to a first-order system */

2026-10-10T17:47:26Z

Reduction to a first-order system

← Older revision		Revision as of 17:47, 10 October 2026
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	== Reduction to a first-order system ==		== Reduction to a first-order system ==

	~~Labelling the~~ successive derivatives of the unknown as new ~~unknowns turns an equation of~~ order ~~$n$ into a system~~ of ~~$n$ equations of order 1~~:		For any ordinary differential equation, considering successive derivatives of the unknown function as new unknown functions may reduce the order of the original equation without losing any information:

	{{#content:Q1991}}		{{#content:Q1991}}

	~~The~~ pendulum ~~illustrates the reduction. Its angle $\theta(t)$~~ obeys the equation of motion		'''Example:'''

			A pendulum obeys the equation of motion

	{{#content:Q1589}}		{{#content:Q1589}}

	which involves the second derivative~~; taking~~ the angular velocity $v=\~~dot~~{\theta}$ as ~~the~~ second unknown ~~gives the equivalent first-order system~~		which involves the second derivative.

			To reduce the equation to the first degree, introduce the angular velocity $w=\frac{d\theta}{dt}$ as a second unknown:

	{{#content:Q1617}}		{{#content:Q1617}}

	No information is lost in the reduction, so the geometric, numerical, and qualitative methods available for first-order equations apply at every order ([[Euler's method]], [[Qualitative methods]]).<ref>{{#cite:Q1576}}</ref>

	== Linear equations and superposition ==		== Linear equations and superposition ==

Rongzhou: /* General form */

2026-10-10T17:36:01Z

General form

← Older revision		Revision as of 17:36, 10 October 2026
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	== General form ==		== General form ==

	~~Solved for the highest derivative, an~~ ordinary differential equation of order $n$ ~~reads~~		An ordinary differential equation of order $n$ has the general form

	{{#content:Q1603}}		{{#content:Q1603}}

	~~and, written implicitly, $F\bigl(x,y,y~~',y''~~,\ldots,y^{(n)}\bigr)=0$. The~~ '''~~order''' $n$ is the order of the highest derivative that appears; the equation of the first order $n=1$ is treated in [[1st order ODE]].~~		'''Example:'''

	A mass $m$ attached to a spring of stiffness $k$, displaced from its rest position and released, obeys the second-order equation		A mass $m$ attached to a spring of stiffness $k$, displaced from its rest position and released, obeys the second-order equation